Monty Hall Problem Simulator
Simulate the famous Monty Hall probability puzzle.
See why switching doors wins 2/3 of the time through hands-on simulation.
The Monty Hall Problem Named after Monty Hall, the host of the American TV game show “Let’s Make a Deal” (which first aired in 1963 in the United States), this probability puzzle has confused mathematicians, professors, and the general public for decades.
The Setup You are on a game show. There are 3 doors. Behind one door is a car (the prize). Behind the other two doors are goats (no prize). You pick a door. The host, who knows what is behind each door, opens a different door revealing a goat. The host then asks: “Do you want to switch to the other unopened door?”
The Counterintuitive Answer You should ALWAYS switch. Switching wins 2/3 of the time (66.7%). Staying wins only 1/3 of the time (33.3%). Most people’s intuition says it should be 50/50, but this is wrong.
Why Switching Wins When you first pick a door, there is a 1/3 chance you picked the car and a 2/3 chance you picked a goat. The host ALWAYS reveals a goat from the remaining doors (he never reveals the car). If you originally picked a goat (2/3 chance), switching takes you to the car. If you originally picked the car (1/3 chance), switching takes you to a goat. So switching wins 2/3 of the time.
The Key Insight The host’s action is NOT random. He always reveals a goat. This means the host’s reveal gives you information. The 2/3 probability that the car was behind one of the two doors you did NOT pick is now concentrated entirely on the one remaining door.
Historical Controversy When Marilyn vos Savant published the correct answer in her Parade magazine column in 1990, she received thousands of angry letters — including from PhD mathematicians — insisting she was wrong. Computer simulations eventually proved her right. It remains one of the most famous examples of how human intuition fails with conditional probability.
This Simulator Run hundreds or thousands of simulated games to see the statistics for yourself. The law of large numbers means the more games you simulate, the closer the results will converge to the true 1/3 vs 2/3 probabilities.